Of course, some of them must be copies, but it seems to me that most of our knowledge about the lifetimes of the N.T. persons comes from the N.T. manuscripts, so are we just "taking their word for it" that they are describing events that happened at least 100 years prior, or am I missing some facts?

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Google "dating of biblical texts" to obtain a plethora of references to the techniques used. As different combinations of these techniques are used for various books and texts, this question is seems too broad to be properly addressed. Is there a particular text you are concerned with?
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Pieter GeerkensAug 17 '13 at 20:40

kojiro, As @PieterGeerkens pointed out reasons differ for different manuscripts in question. In the current format, however, this question will not get good answers here. Most answers will be too long and would add hardly anything that wikipedia does not cover. I respect your question very much and would request that you post it for some specific manuscripts so that we can answer it in a concise manner. Just to be clear, this question is not off topic. It is just too broad.
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Apoorv KhurasiaAug 18 '13 at 12:42

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Forensic analysis, and with respect to ancient documents, particularly palaeography, the scientific examination of ancient documents - reveals that none of manuscripts we have are old enough to be autographs, based on our knowledge of when the actual events occurred.
See: Rylands Library Papyrus P52.

What more direct evidence do you think we could have, being almost 2000 years after the actual events? All copies signed with names of the scribes who copied the material? But that is not the case. That someone attested to destroying all the originals?

As with all of our studies of ancient texts, artifacts, ruins, mounds, etc, we determine their nature by comparing our forensic analysis of the artifact itself with the archaeological, historical and anthropological record and context in which they were found and refer to. That method represents our "state of the art", and our advances in this field are simply advances in our ability to do more accurate forensic analysis and contextual juxtaposition.

That is really the best we can ever hope to do (at least until we manage to perfect safe time travel...) when dealing with material from the remote past, barring exceptional cases where we are lucky enough to happen upon something that yields direct first hand information. In the case of NT, I think it is safe to say that we have no such first hand information, or it would be headline material.

There's not really any evidence that there ever were such a thing as "originals" anyway. It would be fairly easy to argue that they are just compilations of stories about Jesus, which originally got added to over time, much like a snowball rolling downhill, gathering up any material that sticks well as it goes.
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T.E.D.♦Aug 19 '13 at 20:09

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@T.E.D. I don't know the particulars on the NT, but usually in such cases, at some point someone sits down and collects-edits the compilations into a discreet volume with a title like "The Gospel of John" or whatever - that would be considered an original in this case - that 'anthology' as it were.
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user2590Aug 19 '13 at 22:56